SOME REMARKS ON SELF-POINTS ON ELLIPTIC CURVES
byChristophe Delaunay & Christian Wuthrich

Abstract. — In a previous article ([DW09]), we studied self-points on elliptic curvesof prime conductors, p. The non-triviality of these points was proved in general using
a local argument and the modular parametrization over the field Qp . In this paper, we
focus on the special case of Neumann-Setzer curves and we give an alternative proof of
the non-triviality of self-points using the complex side of the modular parametrization.
To obtain this, we prove several estimates, which can be used further to get results
about Neumann-Setzer curves and their modularity.
Dans un article précédent ([DW09]), nous avons étudié les self-points des courbes
elliptiques de conducteurs premiers, p. La non trivialité de ces points a été établie
en général en utilisant un argument local et la paramétrisation modulaire sur lecorps Qp . Dans ce papier, nous nous concentrons sur le cas particulier des courbes
de Neumann-Setzer et nous donnons une démonstration différente de la non-trivialitédes self-points grâce à l’aspect complexe de la paramétrisation modulaire. Pour cela,
nous obtenons plusieurs estimations que nous utilisons ensuite pour prouver d’autresrésultats sur les courbes de Neumann-Setzer et sur leurs aspects modulaires.

1. Introduction
Let E be an elliptic curve defined over Q of conductor N . We denote by X0 (N )the modular curve of level N , it is well known, from the modularity properties of E,
that there exists a modular parametrization:
ϕ : X0 (N )Esending the cusp ∞ ∈ X0 (N ) to the neutral element O of E. A non-cuspidal pointy ∈ X0 (N ) can be understood as an isomorphism class of pairs (F, C) where F is anelliptic curve and C is a cyclic subgroup of order N of F . It is a classical and naturalproblem to study miscellaneous properties of the points x = ϕ(y) ∈ E whenever yhave some specific and “interesting” description in X0 (N ).
For instance, if y is a cusp in X0 (N ), the theory of Manin-Drinfeld gives that ϕ(y)is a torsion-point in E and that its order can be controlled. Modular symbols also
allow us to compute the point ϕ(y) in this case ([Cre97, chapter 2]).

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CHRISTOPHE DELAUNAY & CHRISTIAN WUTHRICH

An other example is given by Heegner points. An Heegner point has the formy = (F, C) ∈ X0 (N ), where F and F/C have complex multiplication by the sameorder O of an imaginary quadratic field. The Gross-Zagier theorem ([GZ86]) gives
necessary and sufficient conditions that x = ϕ(y) is a non-torsion point in E(H)where H is some number field associated to O. The Gross-Zagier theorem has many
theoretical and explicit applications. In particular, in combination with the complex
interpretation of Heegner points, it leads to a very efficient algorithm for computing
a generator of the Mordell-Weil group E(Q) whenever E has analytic rank 1 (forexample, [Coh07, chapter 8]).
The images of some other natural points have also been considered (see [Har79],[Kur73], etc.) with the perspective to study the rank of the E(L) in some infinite
Iwasawa extensions L. A special case of these points are the so-called “self-points” inthe title. They were defined and have also been investigated in [DW09] and [Wut09].
Definition 1. — A self-point, PC ∈ E, is a point PC = ϕ(yC ) where yC is of theform yC = (E, C) ∈ X0 (N ).
Note that there are #P2 (Z/N Z) cyclic subgroups, C ⊂ E, of order N . The questionof the rank generated by the self-points (and also by the “higher” self-points) has beenstudied in generality in [Wut09]. One of the key ingredient is to determine when the
point PC is a non-torsion point. This can indeed be done in most of the cases byconsidering the modular parametrization over the local field Qp where p is some wellchosen prime dividing N .
Whenever the conductor N = p is prime, the local argument is always valid and it
can be shown ([DW09]) that the self-points PC are non-torsion points in E(Q(C)),
where Q(C) is the field of definition of C. This implies that the points (PC )C , where
C is running through the p + 1 cyclic subgroups of order p, generate a group of rank p
in E(K) where K is the compositum of the fields Q(C). Note that the Galois groupof K/Q is G ' PGL2 (Z/pZ).
The aim of this paper is to focus on the special cases when E are NeumannSetzer curves and to show that by considering the modular parametrization ϕ overthe complex field C (rather than Qp ) may also provide some results on these points.
In section 2, we will briefly sum up the results in [DW09] about self-points on
elliptic curves of prime conductor.
Neumann-Setzer curves are special curves of prime conductor p and will be described in section 3.
Then, we will restrict our attention to these curves. In section 4, we will give aprecise description of the modular parametrization over C. This will allow us to study
the map ϕ. In particular, we will obtain an alternative proof that the self-points are
non-torsion.
This requires some technical and more or less precise estimates. We will also usethem in order to give additional remarks that are not exactly related to the studyof self-points but that we believe to be interesting nonetheless. In section 5, we will
study the growth of the modular degree of the Neumann-Setzer curves and give an
explicit way for computing the analytic order of the Tate-Shafarevich groups of the

SOME REMARKS ON SELF-POINTS ON ELLIPTIC CURVES

3

Neumann-Setzer curves.

Acknowledgements
The first author is supported by the ANR project no. 07-BLAN-0248 ”ALGOL”. He
is also a member of the project ”TraSecure” financed by the ”Région Rhônes-Alpes”
in France.

2. Self-points on elliptic curves of prime conductor
Let E be an elliptic curve defined over Q with prime conductor p. We give heresome basic properties related to the self-points of E, see [DW09] for proofs and moredetails.
The number field Q(E[p]) obtained by adjoining the coordinates of the p-torsion
points of E to Q is Galois and the Galois group Gal(Q(E[p])/Q) is identified withGL2 (Fp ) via the classical Galois representation:ρ̄p : Gal(Q(E[p])/Q)

 GL2 (Fp ),which is an isomorphism.
Let C ⊂ E be a cyclic subgroup of order p, then the field Q(C) is the subfield ofQ(E[p]) fixed by a Borel subgroup of GL2 (Fp ) and so it is a primitive non-Galois fieldof degree p + 1. From the fact that Q(C) does not contain any non-trivial subfield,
we can deduce that E(Q(C))tors = E(Q)tors .
As C is running through all the p + 1 cyclic subgroups of order p, the fields Q(C)are all conjugate. Their Galois closure is the field K ⊂ Q(E[p]) and Gal(K/Q) is
identified with PGL2 (Fp ) via ρ̄p . Since the map ϕ is defined over Q, the self-points
inherit of the algebraic properties of Q(C):Proposition 2. — We have:– The point PC lies in E(Q(C)).
– The set {PC }C form a single orbit under the action of Gal(K/Q) in E(K).
It follows immediately from this proposition and from E(Q(C))tors = E(Q)tors that
if a self-point PC were a torsion point it would be rational all the other self-points
should also be rational and equal. This is trivially impossible if deg(ϕ) <Pp + 1 (seeremark 5.1.1 about this fact.). Furthermore, we have that trK/Q PC = C PC is atorsion point, so if PC were rational then PC would be a torsion point.
In [DW09], we proved that this case can not occur since we obtained:
Theorem 3. — With the previous notations, we have:– The self-points PC are of infinite order.
P– The p + 1 self-points {PC }C generate a rank p group in E(K) and C PC is therational torsion point ϕ(0) ∈ E(Q).

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CHRISTOPHE DELAUNAY & CHRISTIAN WUTHRICH

The first point is proved by considering the p-adic interpretation of ϕ. The second point comes from a linear argument (using the irreducibility of the Steinberg
representation of PGL2 (Fp )) and from the first point. This second point is in fact a
corollary of the first one. In section 4, we will give a new proof of the first point using
the complex interpretation of the modular parametrization in the case when E is a
Neumann-Setzer curve.
3. Neumann-Setzer curves
Let u ∈ Z be an odd integer such that u ≡ 3 (mod 4). Suppose that p = u2 + 64
is prime. Such a prime will be called a Neumann-Setzer prime. The Neumann-Setzer
curves of conductor p are the following two isogenous curves:
u+1 2
x + 4x − u
Ep : y 2 + xy = x3 −
4
u+1 2
x −x
Fp : y 2 + xy = x3 −4
We will write E and F if the Neumann-Setzer prime is understood. The curves E
and F are isogenous by an isogeny of degree 2.
The discriminant of E is ∆ = −p2 and its j-invariant is −(u2 − 192)3 /p2 . The
2-division polynomial of E is given by:
P (x) = (4x − u)(x2 + 4)
The group E(Q) contains a rational 2-torsion point
which is given by (u/4, √−u/8).√ The√
two other points of order 2 are defined over Q( −1) and are the points ±(2 −1, −1).
The discriminant of F is ∆ = p and its j-invariant is (u2 + 48)3 /p2 . The 2-division
polynomial of F is given by:u
P (x) = 4x x2 + x − 1 .
4
The points of order 2 of F are:


√
√ 
√
√ 
u+ p u+ p
u− p u− p(0, 0),
,−
and
,−
.
8
16
8
16
√
So that in this case, Q(E[2]) = Q( p).
The curves of prime conductor and, in particular, Neumann-Setzer curves havebeen studied by many authors: [Miy73], [Neu71], [Neu73], [Set75], [SW04],... .
From these sources and from [AU96], we have the following theorem (see [DW09],
for details):
Theorem 4. — Let p = u2 + 64 be a Neumann-Setzer prime with u ≡ 3 (mod 4).
Let E and F be the Neumann-Setzer curves as above.
– We have E(Q) ' F (Q) ' Z/2Z.
– We have X(E/Q)[2] ' X(F/Q)[2] ' {0}.
– The local Tamagawa number of E and F at the prime p is cp = 2.

SOME REMARKS ON SELF-POINTS ON ELLIPTIC CURVES

5– The curve E is the strong Weil curve in its isogeny class and the Manin constantof E is equal to 1.
– The modular degree of E is even if and only if u ≡ −1 (mod 8).
Furthermore, if G is an elliptic curve of prime conductor such that G(Q)tors ' Z/2Z
then G is a Neumann-Setzer curve.
If E is an elliptic curve of prime conductor which is not a Neumann-Setzer curvethen E has conductor 11, 17, 19, 37 or E(Q)tors is trivial. In the last case, E is the
only curve in its isogeny class (and note that its rank can be positive).
It is not known if there exist infinitely many elliptic curves of prime conductor.
Nevertheless, classical conjectures can be applied for the number of Neumann-Setzer
primes.
Conjecture 1. — Let πN S (x) be the number of Neumann-Setzer prime p 6 x and
let:


χ(p)
1 Y
1−
C=
2 p primep−1
where χ(·) = −4is the primitive Dirichlet character modulo 4. As x
 ∞, we
·
have:
Z √x
dt
πN S (x) ∼ C
.
log t
2
The infinite product defining C is converging but is not absolutely convergent since:
1−

χ(p)
χ(p)
∼1−
p−1
p

The number C is the Hardy-Littlewood constant of the polynomial x2 + 64 (this is
the same Hardy-Littlewood constant as for the polynomial x2 + 1). One can find in[Coh] how to compute such constants numerically. In particular, we have:C ≈ 0.686406731409123004556096348363509434089166546754.
Of √course, the conjecture above implies the less precise conjectural estimate πN S (x) ∼2C x/ log x.
As expected, the comparison of the conjectural estimate with the exact values ofthe function πN S (x) for small x is quite convincing.
x
πN S (x)
Z √x
C
2

dt
log t

106
119
≈ 121.19

1012
53996

1018
34898579≈ 53969.76 ≈ 34903256.44

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CHRISTOPHE DELAUNAY & CHRISTIAN WUTHRICH

4. Complex side of Neumann-Setzer curves
We give a description of the analytic point of view of the modular parametrization.
We assume here that p = u2 + 64 is a Neumann-Setzer prime with u ≡ 3 (mod 4).
We consider the Neumann-Setzer curve E = Ep as in the previous section:
u+1 2
x + 4x − u.
4

E : y 2 + xy = x3 −4.1. Complex points of E. — It is well known that there exists an analytic
isomorphism between the complex points E(C) of E and C/Λ where
ω2
Λ=Z
⊕Z
ω1
is the period lattice associated to E. This isomorphism is expressed with the Weierstrass function and its derivative; we denote it by ℘ so that:℘ : C/Λ

∼ E(C).

Following [Coh93, chapter 7], the numbers ω1 and ω2 can be given by:
(1)

ω1 =

(2)

ω2 =

2π
 r 
 ∈ R
p1/4 agm 1, 12 1 + √up

ω1
+i·
2

π
 r 

p1/4 agm 1, 12 1 − √up

Here agm(·, ·) denotes the classical arithmetic-geometric mean.
4.2. Complex L-function of E. — We denote by (an )n>1 the coefficients of the
L-function of E:
X
L(E, s) =an n−s , for <(s) > 3/2.
n>1

It is easy to see that the curve E has split multiplicative reduction at p hence ap = 1.
From the modularity of E, the function L(E, s) has an analytic continuation to thewhole complex plane and satisfies a functional equation. The sign of this functional
equation is ap = +1 so we have: √ sp
Λ(E, s) :=Γ(s)L(E, s) = Λ(E, 2 − s).
2πPFurthermore, the function f (τ ) = n>1 an q n with q = e2iπτ is a newform of weight
2 on Γ0 (p). From the theory of Atkin-Lehner, we have:f (Wp τ ) = −pτ 2 f (τ )where Wp =

0
p

−1
0


is the Fricke involution.

SOME REMARKS ON SELF-POINTS ON ELLIPTIC CURVES

7

4.3. Complex modular parametrization of E. — Let H be the upper halfplane, H = {τ ∈ C, =(τ ) > 0}. The complex points of the space X0 (p) can beinterpreted as the quotient of H ∪ Q ∪ {i∞} by the congruences subgroup Γ0 (p). Thenthe modular parametrization of E factorizes as ϕ = ℘ ◦ φ:ϕ

X0 (p)

 E(C)


φ

℘


C/Λ
where φ is given by the following converging series for τ ∈ X0 (p) \ {cusps}:φ :X0 (p)
τ

 C/Λ
X an

e2iπnτ
nn>1

The image of the cusp 0 ∈ X0 (p) is a rational torsion point, hence ϕ(0) ∈ E(Q)tors .
In fact, we have:
X an
√
ω1
e−2πn p ∈ Z .
φ(0) = L(E, 1) = 2
n
2
n>1

So ϕ(0) = k u4 , − u8 with k = 0 or k = 1.
Proposition 5. — If we assume the truth of Birch and Swinnerton-Dyer conjecture
for E thenu u
ϕ(0) =,−.
4
8
Proof. — Indeed, if we assume the Birch and Swinnerton-Dyer conjecture is valid,then we have:ω1 · cpω1
L(E, 1) =|X(E/Q)| =|X(E/Q)|
2|E(Q)tors |2
The result follows from the fact the |X(E/Q)| is odd (if finite) by a theorem of Steinand Watkins [SW04].
The index of Γ0 (p) in SL2 (Z) is p + 1. As a set of representative of SL2 (Z) modulo
Γ0 (p) we take the matrices:




1 0
0 −1
S0 =
and Sj =for j = 1, 2, . . . , p.
0 1
1 j
Let τ0 ∈ H so that j(τ0 ) is the j-invariant of E, then the analytic interpretation of
the self-points (E, C) are the p + 1 points τ0 , τ1 ,...,τp with:
τj = Sj τ0 =−1
for j = 1, 2, . . . , p.
τ +j

8

CHRISTOPHE DELAUNAY & CHRISTIAN WUTHRICH

The Hecke operator Tp acts on modular forms of weight 2 on Γ0 (p), by definition we
have:


p
1X
τ +j
f
.
Tp f (τ ) =p j=1
pSince the function f is a Hecke eigenform with eigenvalue ap = 1, we also have
Tp f (τ ) = ap f (τ ) = f (τ ). Taking the primitive, we deduce:


p
X
τ +j
ϕ(τ ) =
ϕ.
p
j=1
The constant term in the integration is 0 as it can been seen taking τ = i∞. Furthermore, f is also an eigenform of the Fricke involution so, ϕ ◦ Wp (τ ) = −ϕ(τ ) + ϕ(0).
Hence:



p
X
τ +j
+ pϕ(0)ϕ(τ ) = −
ϕ Wp
p
j=1


= Sj τ . So, we have:
Remark that in any case pϕ(0) = ϕ(0) and that Wp τ +jp
Proposition 6. — For τ ∈ X0 (p), we have:ϕ(τ ) +p
X

ϕ(Sj τ ) = ϕ(0).

j=1

In particular, if we take τ = τ0 , we obtain an other approach in the proof of thesecond point of theorem 3:
Corollary 7. — We have:X

PC = ϕ(0).

C

Note that the proof we have just given for the trK/Q PC is clearly analytic comparedto the one given in[DW09]).
4.4. Estimates for ω1 and ω2 . — Recall that p = u2 + 64 with u ≡ 3 (mod 4).
Hence, we have p > 73 and:
r
 
|u|
64
32
1
=1−
+O
.
√ = 1−pppp
We define x+ and x− by:x+

and x−

s 

|u|
1
1+
=
2
p
s 

1
|u|
=
1−
.
2
p

SOME REMARKS ON SELF-POINTS ON ELLIPTIC CURVES

9

Proposition 8. — With the notations above (in particular p > 73), we have:
1−

7
6 agm (1, x+ ) 6 1
p

Proof. — Let x = x+ , we clearly have that x 6 agm(1, x) 6 21 (x + 1). We obtain the
proposition by a straight forward study of x in function of p > 73.
For estimating agm(1, x− ), we will need the following lemma:
Lemma 9. — For x ∈]0, 1], we have:− log x +3π1
5log 2 6
6 − log x + log 22
2 agm(1, x)
2

Proof. — We let
Z π/2g(x) =

dtp

cos2 t + x2 sin2 t
1
so that we have agm(1, x) = π2 · g(x). The change of variables t0 = cos t/ sin t gives
Z ∞dtpg(x) =.
2 + 1)(t2 + x2 )(t0R∞R √x R ∞Now, we split the integral 0 as the sum 0 + √x . The change of variables t0 = x/t,
for x > 0, in the latter integral shows that we have
Z √x
dt
p
g(x) = 2.
2(t + 1)(t2 + x2 )00This gives us the inequalities
Z √x
Z √x
dt
dt
2
√
√
√
6 g(x) 6 2
2
2
2
1+x 0
t +x
t + x20
R √x dt√
and, since we have 2 0 √t2 +x2 = − log x + 2 log(1 + 1 + x), the lemma follows
from
√
√

1
− log x + 2 log(1 + 1 + x) √6 g(x) 6 − log x + 2 log(1 + 1 + x)
1+x
and from a study of the functions on the right and on the left of this inequality.
Proposition 10. — We have:


1
8
1
1
log p + log
6
6 (log p + log 2)) .
π
25
agm(1, x− )
π
 

p
Proof. — We have log x− = 21 log 12 1 − 1 − 64/p and an easy calculation proves
that, for p > 73:
 25
p
1
16
6
1 − 1 − 64/p 6
.
p
2
p

10

CHRISTOPHE DELAUNAY & CHRISTIAN WUTHRICH

Hence,
1
25
1
16
− log
6 − log x− 6 − log .
2p2pThen, we use lemma 9 to conclude.
Corollary 11. — Let p = u2 + 64 be a Neumann-Setzer prime with u ≡ 3 (mod 4).
If u > 0, we have:
1
2π
1
2π
.
4 6 ω1 6 1/4
1/4
p 1− pp 1 − p7
Whereas, if u < 0:
2
p1/4


log p + log

825


6 ω1 62(log p + log 2)
p1/4

Proof. — We apply the two propositions above with the definitions of ω1 and ω2 (seeequations (1) and (2)).
Let remark that in any case (u > 0 or u < 0):
(3)

ω1 >

2π
p1/4

4.5. Proof that the self-points are non-torsion. — We can now prove that theself points are non-torsion using the complex modular parametrization.
It follows from E(Q(C))tors = E(Q)tors that if PC were a torsion point then it would
be a rational torsion point. So, in order to prove that PC is not a torsion point it issufficient to prove that φ(τ0 ) 6= 0 (mod 12 Λ) for a certain τ0 ∈ H such that j(τ0 ) is
the j-invariant of E.
For that we let τ0 = ω2 /ω1 if u > 0 and τ0 = (ω2 − ω1 )/(2ω2 − ω1 ) if u < 0. In
fact, for u < 0, we have:


ω2
1 −1
τ0 =
.
2 −1
ω1
Since the matrix above belongs to SL2 (Z), we see that in each case the image of
yC = (E, C) ∈ X0 (p) in C/Λ (for a certain C depending on the sign of u) is givenby φ(τ0 ). It is easy to see that we have:τ0 =1
agm(1, x+ )
+i·
∈ H.
2
2 agm(1, x− )Hence, φ(τ0 ) is real and we just need to prove that φ(τ0 ) 6= 0 (mod ω21 ).
Theorem 12. — With the notations above, we have the following estimates:−

6
1
6 φ(τ0 ) 6 −.
p
10p

In particular, φ(τ0 ) 6= 0, ω1 /2 (mod ω1 ) and PC is a non-torsion point.

SOME REMARKS ON SELF-POINTS ON ELLIPTIC CURVES

11

Proof. — We let t = =(τ0 ) and q = e2iπτ0 = −e−2πt . From propositions 8 and 10, we
have:


1
7
8p
1
1−
log
6t6
log 2p2πp252πHence,
 1−7/p
1
25
−2πt
.
6e
6
2p
8p
 1−7/p
25
But, we have 8p
6 5/p whenever p > 73. Finally, we obtain that q < 0 and:1
5
6 |q| 6 .
2p
p

(4)

We have ann 6 1 (see [GJP+ 09]) so φ(τ0 ) = q + ε where:
(5)

|ε| 6

X
n>2

|q|2 =|q|2
25
6.
1 − |q|p(p − 5)

Then (4) and (5) give the inequality for φ(τ0 ). We deduce that φ(τ0 ) 6= 0 and that|φ(τ0 )| < π/p1/4 6 ω1 /2 by equation (3).
In fact, we have proved that:φ(τ0 ) = q + O 1/p2



where the constant is absolute and where q  p1 . We will use that in the next section.
4.6. Isogeneous self-points. — Let F be the Neumann-Setzer curve that is isogeneous to E (see section 3). Then, one can consider the point zD = (F, D) ∈ X0 (p)where D ⊂ F is a sub-group of order p. The image QD = ϕ(zD ) are also interesting
point in E(K). We can use exactly the same method as before to proof that QD is a
non-torsion point for all D. The question of the independence of the p + 1 points QD
and the p + 1 points PC is natural. We believe that the only relations between thosepoints are given by:
trK/Q PC = trK/Q QD = ϕ(0)This would follow from the fact that if C and D are chosen so that they are definedover the same field Q(C) then the points PC and QD are independent in E(Q(C))(see [DW09]).
Let t0 be as in the previous section so that t0 correspond to
1(E, C) = C/Zτ0 ⊕ Z, h ipwhere h p1 i denotes the cyclic sub-group of order p generated by 1/p (mod Zτ0 ⊕ Z).
For the curve F , we can choose the point τ00 = 2τ0 corresponding to:
1(F, D) = C/Zτ00 ⊕ Z, h ip12CHRISTOPHE DELAUNAY & CHRISTIAN WUTHRICH

where h p1 i denotes here the cyclic sub-group of order p generated by the point 1/p(mod Zτ00 ⊕ Z). The 2-isogeny between F and E correspond to the map induced by
the identity:
C/Zτ00 ⊕ Z
x(mod Zτ00 ⊕ Z)

 C/Zτ0 ⊕ Z
 x (mod Zτ0 ⊕ Z)

since Zτ00 ⊕ Z ⊂ Zτ0 ⊕ Z. This 2-isogeny being rational, the point (E, C) and (F, D)
are defined over the same field, E(Q(C)) = E(Q(D)).
0
We have q 0 = e2πτ0 = q 2 with q = e2iπτ0 from the previous section. Using the same
technique as in the previous section, we prove that:
 
1
0
2
φ(τ0 ) = q + O
p4
where the implied constant is absolute.
We believe that the points PC and QD are linearly independent in E(Q(C)) but weare not able to prove it. Nevertheless, if there exists a linear relationship of thesepoints, it would involve rather large coefficients since:
Theorem 13. — With the notations above, suppose that there exist `, m ∈ Z \ {0}such that:`PC + mQD ∈ E(Q(C))tors
then max(|`|, |m|)  p3/4 , where the implied constant is absolute.
Proof. — Recall that E(Q(C))tors = E(Q)tors , hence if `PC + mQD ∈ E(Q(C))tors , wewould have:ω1`φ(τ0 ) + mφ(τ00 ) ∈Z.
2If `φ(τ0 ) + mφ(τ00 ) = 0 ∈ C then ` 6= 0 since φ(τ00 ) does not correspond to a torsion
point and so:|m||φ(τ0 )|=p ,|`||φ(τ00 )|and |m|  p.
If `φ(τ0 ) + mφ(τ00 ) = λω1 /2 for some λ ∈ Z \ {0} then in this case:
1
1
ω1
2π
|`| + |m| 2  |`φ(τ0 ) + mφ(τ00 )| >
> 1/4
p
p
2
p
so |`|  p3/4 or m  p7/8 .

SOME REMARKS ON SELF-POINTS ON ELLIPTIC CURVES

13

5. Some other consequences
5.1. Growth of the modular degree of Neumann-Setzer curves. — Thereis an interesting consequence about the degree of the modular parametrization of the
optimal Neumann-Setzer curve E. Indeed, using our estimates for ω1 and ω2 , it is
not difficult to see that we have:
2π log(p) ∞.
vol(E) = ω1 · =(ω2 ) ∼as p = u2 + 64√p
Furthermore, since the Neumann-Setzer curves are semi-stable, the modular degree
is given by the following formula.
deg(ϕ) =p L(Sym2 f, 2).
2π vol(E)Where L(Sym2 f, s) is the symmetric-square L-function associated to L(E, s) normalized so that s = 3/2 is the point of symmetry in the functional equation (let’s remarkthat the conductor being square-free, there is no difference between the primitive andthe imprimitive symmetric-square).
Using the classical upper bound L(Sym2 f, 2)  log(p)3 and the deeper lower
bound L(Sym2 f, 2)  1/ log(p) obtained by Goldfeld, Hoffstein and Lieman [HL94],
we deduce:
Theorem 14. — Suppose that there are infinitely many Neumann-Setzer primes p ∞:then for the Neumann-Setzer curves E of conductor p we have as pdeg(ϕ)  p3/2 log(p)2
deg(ϕ)  p3/2 / log(p)2Note that the degree of the modular parametrization is “large” because the jinvariants tend to infinity with p. Indeed, in [Del03]), it is proved that:
Theorem 15. — Let G be an infinite family of semi-stable elliptic curves G definedover Q with conductor NG such that the j-invariant of G is bounded for all G ∈ G
and such that G are the strong Weil curves in their isogeny classes. If ϕG denotes
the modular parametrization of G, then as NG
 ∞:
7/6

deg(ϕG )  NG (log NG )3
7/6

deg(ϕG )  NG / log NG
In this context, the power 3/2 of theorem 14 should be compared with the power
3/2 in the previous estimates.
5.1.1. Remark. — In fact, Watkins [Wat04] gave a very explicit version of the lower
bound L(Sym2 f, 2)  1/ log p. (He normalized L(Sym2 f, s) so that 1/2 is the pointof symmetry.) Using his work and our estimates we have for a Neumann-Setzer curve
E of conductor p:p3/2
deg(ϕ) > 0.0006 ·.
log(p)2

14

CHRISTOPHE DELAUNAY & CHRISTIAN WUTHRICH

Hence the inequality deg(ϕ) < p+1 never occurs for p > 1.8·1012 (but there probably
exists a much smaller value of B such that deg(ϕ) > p + 1 for all p > B).
5.2. Explicit computations of the analytic order of the Tate-Shafarevichgroups of Neumann-Setzer curves. — Throughout this section, we assume thetruth of the Birch and Swinnerton-Dyer conjecture for E. So that we have:|X(E)| =(6)

2L(E, 1)
ω1

Suppose that we want to compute numerically the values of |X(E)|. Then, we haveto compute numerically the series:
X an√
L(E, 1) = 2
e−2πn/ p .
nn>1

And we need to truncate the series using sufficiently many coefficient. That meansthat we write:
X an
X an
√
√
(7)
e−2πn/ p =e−2πn/ p + Error
n
n
n>1

n<N0

and we need to take N0 sufficiently large to be are able to recognize |X(E)| fromequation (6).
Theorem 16. — Let η > 0, there exists an explicit K > 0 such that for all NeumannSetzer curves of conductor p > K it is sufficient to take1 + η√
N0 >p log p4π
in equation (7) in order to determine |X(E)|.
Proof. — Let η > 0, it is well known that there is an absolute constant K0 such that|an | 6 n1/2+η for all n > K0 (this is in fact true for all elliptic curves defined over Q;
note that for η =P
1/2 we can take √K0 = 1). P√
We write S = n<N0 ann e−2πn/ p and ε = n>N0 ann e−2πn/ p . Hence we have:11(4S − 4|ε|) 6 |X(E)| 6(4S + 4|ε|).
ω1
ω1
Since |X(E)| is an odd square, the length of the interval in the inequality above has
to be less than 4 in order to determine |X(E)|. So, we need |ε| < ω1 /2 hence, weneed:πε < 1/4 .
p√√
Suppose that p > K0 then N0 > p and we have:
X |an |
X√√
1
−2πn/ p|ε| 6e−2πn/ p 6enn1/2−η
n>N0
n>N0
√ N0
e−2π/ p1
1 p1/2  −2π/√p N0√  6 1−2η
6
e
.
1−2η
4
1 − e−2π/ p
p 4
p 4

SOME REMARKS ON SELF-POINTS ON ELLIPTIC CURVES

15(Note that p > 73). The values of N0 is sufficient for the theorem.
√
In particular, taking η = 1/2, we need to take N0 > 81 p log p.
Using this, we have computed several values of |X(E)| of Neumann-Setzer curves.
We give here some numerical investigations related to these values. In particular, weconsider the study of the frequencies of |X(E)| that are divisible by the primes q
for q = 3, 5, 7, · · · (trivially, the case q = 2 is a special one, and there is nothing
to say about it). Computing sufficiently enough values of |X(E)|, we can compare
these numerical frequencies with the heuristics on Tate-Shafarevich groups ([Del01],[Del07]). In this case, the heuristics predict that, if q is (an odd) prime, the frequencyof occurrences of q | |X(E)| should be given by:

Y
1
1
1
1
f (q) = 1 −1 − 2k−1 = + 3 − 5 + · · · .
q
q
q
q
k>1

We first computed the values of |X(E)| for the 53996 Neumann-Setzer curves of conductor p 6 1012 . The largest value |X(E)| = 1232 was obtained for p = 974419714193(u = 987127). It is worth noting that |X(E)| = 1132 occured for u = 984355 (113 isprime). In fact, except for q = 97 and q = 109, all the primes q 6 113 divides |X(E)|for at least one Neumann-Setzer curve E with p 6 1012 .
We obtained the following results for the frequency of occurrences of q | |X(E)|:q
3Frequency of q | |X(E)| 0.353
f (q) ≈ 0.361

5
0.185
0.207

7
0.118
0.145

11
0.056
0.092

Expect for p = 3, the numerical values are not so close than the expected ones.
Indeed, we believe that Tate-Shafarevich groups acquire their expected behavior for
rather large conductor. To illustrate this, we also computed the orders of 10000 TateShafarevich groups of the first Neumann-Setzer curves E having conductor p > 1015 .
We obtained the following table:
q
3
5
7
11
Frequency of q | |X(E)| 0.368 0.198 0.140 0.084
We should mention that the largest value is |X(E)| = 2992 and that the averagevalue for these 10000 values of |X| is ≈ 1378.

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[Coh07]

February 26, 2010
Christophe Delaunay, Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan,
43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France.
E-mail : delaunay@math.univ-lyon1.fr • Url : http://math.univ-lyon1.fr/∼delaunay
Christian Wuthrich, Shool of Mathematical Sciences, University of Nottingham, Nottingham
NG7 2RD, United Kingdom. • E-mail : christian.wuthrich@nottingham.ac.uk